The Expansibility Factor Equations in ISO and ISO : Do They Deliver What They Promise?
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1 The Expansibility Factor Equations in ISO and ISO 567-4: Do They Deliver What They Promise? Michael Reader-Harris, NEL INTRODUCTION The expansibility factor equations in ISO 567-2:2003 [] and ISO 567-4:2003 [2] are very different. The equation for orifice plates was derived from experimental data. It will be expected that it will work for the original orifice plates whose data were used to derive it. The crucial question is whether it works for orifice plates from which data were not used to derive the equation. In contrast to the expansibility factor equation for orifice plates that for Venturi tubes given in Section 5.6 of ISO 567-4:2003, is based on theory alone. However, experience has shown that it leads to bias in measurements. In this paper, data for β = 0.4, where β is the diameter ratio (the ratio of the orifice diameter, d, to the pipe internal diameter, D), have been analysed. 2 ORIFICE PLATES The equation for the orifice plate expansibility factor, ε, in ISO 567-:99 [3] was the equation derived by Buckingham [4] largely on the basis of data collected at tests in Los Angeles in 929: ε 4 p = ( β ), () κp where p is the differential pressure, p the (absolute) static pressure at the upstream pressure tapping and κ is the isentropic exponent. An improved expansibility factor equation [5] was derived and included in ISO 567-2:2003: 4 8 p ( ) 2 κ ε = + β + β, p where p 2 is the (absolute) static pressure at the downstream pressure tapping. The uncertainty of Equation (2) is stated [] to be p 3.5 κp per cent. (3) ISO/TC 30/SC 2 decided to include Equation (2) in ISO 567-2:2003; API decided to collect additional expansibility-factor data in a wider range of pipe sizes: from each set of data the value of b* in p κ ε = b * 2 (4) p (2)
2 was determined. Those values of b* calculated from orifice plates whose edges were sharp and which were installed with a long upstream length are shown in Figure : they have been taken from Hodges [6]. The new CEESI data were taken in 2, 3, 4, 6 and 0 pipes; the SwRI data in this figure were taken in a 4 pipe. The SwRI tests are described by Morrow [7,8]. The SwRI data were calculated assuming a value of.3 for κ; George [9] (2008) states that using the true value of κ would have a negligible effect on the plotted data. Data taken with a flow conditioner 7D or 0D upstream of an orifice plate are not included in Figure (they are described below and included in Figure 2). By way of comparison those original data (i.e. the data from which Equation (2) was derived) for which Reynolds number corrections were not required are also shown in Figure. The key test for the expansibility-factor equation in ISO 567 (as for the discharge-coefficient equation) is that it works for orifice plates to which it was not originally fitted: it can be seen that all the new points lie within the uncertainty band for the equation in ISO 567-2: b* β 4 Figure Values of b* in Equation (4) NEL Gaz de France CEESI New CEESI SwRI (long upstream pipe, sharp orifice) ISO 567-: 99 ISO 567-2: 2003 Uncertainty in ISO 567-2:2003 NOTE The uncertainty in ISO 567-2:2003 is a percentage figure and so the absolute values depend slightly on p/(κp ); accordingly the uncertainty curves shown in this figure are approximate. The SwRI data taken with a flow conditioner upstream of the orifice plate are shown in Figure 2. The 6 data were taken in a 2D tube with a CPA 50E Flow Conditioner 0D upstream of the orifice plate; the 3 data were taken in a 7D tube with a CPA 50E Flow Conditioner 7D upstream of the orifice plate. On the basis of the data in Morrow [0] the NOVA (now CPA) 50E did not pass the flow conditioner test in of ISO 567-:2003 for β = 0.75 when the flow conditioner was 7D upstream of the orifice plate. 2
3 b* ": CPA 50E 0D upstream of orifice in 2D tube 3": CPA 50E 7D upstream of orifice in 7D tube ISO 567-: 99 ISO 567-2: 2003 Uncertainty in ISO 567-2: β 4 Figure 2 Values of b* in Equation (4) for SwRI tests downstream of the CPA 50E Flow Conditioner NOTE The uncertainty in ISO 567-2:2003 is a percentage figure and so the absolute values depend slightly on p/(κp ); accordingly the uncertainty curves shown in this figure are approximate. Where the flow conditioner passed the flow conditioner test in 7.4. of ISO 567- :2003 the points in Figure 2 lie within the uncertainty band for the equation in ISO 567-2: VENTURI TUBES Although the best way to collect expansibility-factor data, the method used for orifice plates in 2, is to collect data at constant Reynolds number by installing the differential pressure meter downstream of a choked sonic nozzle and with a valve downstream of the differential pressure meter, the Venturi-tube data analysed here were taken in a recirculating loop and the Reynolds number was not constant. Four sets of data, each taken in air, were analysed: one 2, two 4 and one 6. The 2 and 4 data, as calculated using the expansibility factor equation in ISO 567-4:2003, are plotted in Fig. 7.2 of Reader-Harris []. Some details of each Venturi tube are given in Table. Table The Venturi tubes used in this analysis tapping Max. diff. pressure, p, at each Nominal diameter Number D d diameter upstream static pressure, p mm mm mm p, bar p, bar a M C Maximum p/p
4 Each of the data sets was fitted in the same way: 6 cre D /0 C = a be (5) ε = p 2 p κ a (6) where C is the discharge coefficient and Re D is the pipe Reynolds number. The results were as shown in Table 2. Table 2 The best-fit coefficients of Equations (5) and (6) a b c a M C The variation in predicted values of c is very large. It might be more appropriate to determine c so that, as in Equations (7.8) and (7.9) of Reader-Harris [], neglecting the term in β 4, C Re = a be tap (7) and then to determine a and b by fitting the data, where Re tap is the tapping-hole Reynolds number defined by uτ dtap Re tap =, (8) ν d tap is the tapping diameter, ν is the kinematic viscosity, u τ is the friction velocity, τ w, τw is the wall shear stress and ρ is the density. ρ In a standard Venturi tube in both tapping planes Lindley [2] made measurements of τ w from which the friction factor, λ, can be deduced, since following Schlichting [3], λ is given by w 2 λρ 8 u τ =. (9) where u is the mean velocity. In the throat Lindley s measurements of λ were approximately 8 per cent higher than would be obtained in a straight pipe of the same relative roughness and Reynolds number; so here a figure of 0.05 was used for λ th (for 0 6 < Re and k/d = 0-4,.8λ will always be within 6 per cent of 0.05 according to the Moody Diagram, where k is the uniform equivalent roughness). From Equations (7) (9) the value of c in Equation (5) was calculated, and on fitting the data using Equations (7) and (6) the results were as shown in Table 3. 4
5 Table 3 The best-fit coefficients of Equations (7) and (6) a b c a M C The values of b in Table 3 are quite close to , the value that would have been expected from Equations (7.8) and (7.9) of Reader-Harris [], neglecting the term in β 4. When the data were fitted using Equations (7) and (6) with b = , the results were as shown in Table 4. Table 4 The best-fit coefficients of Equations (7) and (6) with b = a b c a M C Because for the 6 data the maximum value of p/p was only the value of a varies much more between Tables 2, 3 and 4 for the 6 data than for the 2 and 4 data. If the values for the other three meters are averaged the values in Tables 2, 3 and 4 are , and respectively. Just using the first three values from Table 2, since they have a smaller standard deviation than those in Tables 3 or 4, the uncertainty in a is t s = (0) 2 = Using the first three values, the mean value of a in Tables 3 or 4 does not differ from that in Table 2 by more than a quarter of this uncertainty. Therefore, a reasonable expansibility equation for a Venturi tube of diameter ratio 0.4 is p 2 ε = p with an uncertainty of p p p p κ 2 κ () %. (2) κ 5
6 Using this equation changes the measured discharge coefficients for the four Venturi tubes. Their discharge coefficients are shown in Figures 3 to 6. Discharge Coefficient bar g using ISO expansibility 60 bar g using ISO expansibility 20 bar g using Equation () for expansibility 60 bar g using Equation () for expansibility Pipe Reynolds Number Figure 3 The discharge coefficient for 2 Venturi tube bar g using ISO expansibility 60 bar g using ISO expansibility 20 bar g using Equation () for expansibility 60 bar g using Equation () for expansibility Discharge Coefficient Pipe Reynolds Number Figure 4 The discharge coefficient for 4 Venturi tube 28907M 6
7 Discharge coefficient bar g using ISO expansibility 35 bar g using ISO expansibility 60 bar g using ISO expansibility 20 bar g using Equation () for expansibility 35 bar g using Equation () for expansibility 60 bar g using Equation () for expansibility Pipe Reynolds Number Figure 5 The discharge coefficient for 4 Venturi tube 28907C Discharge coefficient bar g using ISO expansibility 45 bar g using ISO expansibility 70 bar g using ISO expansibility 20 bar g using Equation () for expansibility 45 bar g using Equation () for expansibility 70 bar g using Equation () for expansibility Pipe Reynolds Number Figure 6 The discharge coefficient for 6 Venturi tube The improvement in the measured discharge coefficients by changing the expansibility equation is obvious, especially in Figures 3 to 5, based on data which in each case have a wide spread of p/p. 7
8 The difference between the ISO expansibility factor equation and Equation () is approximately equal to the uncertainty of the ISO expansibility factor equation as stated in Section 5.8 of ISO 567-4: CONCLUSIONS Since the orifice plate expansibility factor equation in ISO 567-2:2003 was published new expansibility factor data have been published. These data confirm that the ISO expansibility factor equation does deliver what it promises. Data for β = 0.4 from which a Venturi tube expansibility factor equation can be derived have been analysed. The difference between the ISO expansibility factor equation and the new equation is approximately equal to the uncertainty of the ISO expansibility equation as stated in Section 5.8 of ISO 567-4:2003. It seems that it just fails to meet what it promises. The most important conclusion of this work is to suggest that it would be appropriate to collect data on the Venturi tube expansibility factor over the full range of diameter ratio, so that the expansibility factor equation in ISO 567-4:2003 may be revised. 5 NOTATION D Pipe internal diameter (upstream) d Orifice diameter d tap Tapping diameter k Uniform equivalent roughness p Absolute static pressure at upstream pressure tapping p 2 Absolute static pressure at downstream pressure tapping Re Reynolds number Re D Pipe Reynolds number Re tap Tapping-hole Reynolds number (see Equation (8)) s Standard deviation t n Student s t with n degrees of freedom u Mean velocity u τ Friction velocity (= τ w ρ β Diameter ratio (= d/d) p Differential pressure ε Expansibility (expansion) factor κ Isentropic exponent λ Friction factor ν Kinematic viscosity ρ Density τ w Wall shear stress 6 ACKNOWLEDGMENTS This paper is published by permission of the Managing Director, NEL. 7 REFERENCES [] INTERNATIONAL ORGANIZATION FOR STANDARDIZATION. Measurement of fluid flow by means of pressure differential devices inserted in circular cross-section conduits running full - Part 2: Orifice plates. Geneva: ISO 567-2:
9 [2] INTERNATIONAL ORGANIZATION FOR STANDARDIZATION. Measurement of fluid flow by means of pressure differential devices inserted in circular cross-section conduits running full - Part 4: Venturi tubes. Geneva: ISO 567-4:2003 [3] INTERNATIONAL ORGANIZATION FOR STANDARDIZATION. Measurement of fluid flow by means of pressure differential devices - Part : Orifice plates, nozzles and Venturi tubes inserted in circular cross-section conduits running full. Geneva: ISO 567-:99 [4] BUCKINGHAM, E. Notes on the orifice meter: the expansion factor for gases. Bureau of Standards Journal of Research, Research Paper No 459, 9, July 932 [5] READER-HARRIS, M. J. The equation for the expansibility factor for orifice plates. In: Proc. FLOMEKO 98, Lund, Sweden, 998 [6] HODGES, C. Paper summarizing API investigative testing for existing square edge flange tap orifice expansion factor equations. In: Proc. 6 th Int. Symp. Fluid Flow Meas., Querétaro, Mexico, 2006 [7] MORROW, T. B. Orifice meter expansion factor tests in 4-inch and 6-inch meter tubes. GRI Report GRI-04/0042. Gas Research Institute, Chicago, 2004 [8] MORROW, T. B. Metering research facility program: additional studies of orifice meter installation effects and expansion factor. GRI Report GRI- 04/0246 on SwRI Project No Gas Research Institute, Chicago, 2005 [9] GEORGE, D. L. Revised analysis of orifice meter expansion factor data. Catalog No. L52299 of Pipeline Research Council International Inc, Houston, Texas: Technical Toolboxes Inc, 2008 [0] Morrow T B (997) Orifice meter installation effects: development of a flow conditioner performance test. Report prepared by SwRI as GRI Report No GRI-97/0207, Gas Research Institute, Chicago [] READER-HARRIS, M. J. Orifice plates and Venturi tubes. Springer, 205 [2] LINDLEY, D. Venturimeters and boundary layer effects. PhD Thesis, Cardiff: Dept. of Mech. Eng, Univ. Coll. of South Wales and Monmouthshire, 966 [3] SCHLICHTING, H. Boundary layer theory. McGraw-Hill, New York, 960 9
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